Programming tableaux systems with the LoTREC prover

1
Programming tableaux systems with the LoTREC
prover

• Andreas Herzig
• IRIT-CNRS
• Toulouse
• http//www.irit.fr/Andreas.Herzig/

2
What this is about

• in focus
• the tableaux method
• for logics with possible worlds semantics
• and combinations thereof
• as a computerized proof system LoTREC
• not in focus
• sequent calculi
• efficiency issues

3
Overview

• possible worlds semantics quickstart
• tableaux systems basic ideas
• tableaux systems basic definitions
• tableaux for simple modal logics
• tableaux for transitive modal logics
• tableaux for intuitionistic logic
• tableaux for other nonclassical logics
• tableaux for modal logics with transitive closure
  and other modal and description logics
• tableaux for 1st order logic
• some implemented tableaux theorem provers

4
Possible worlds models

• possible worlds model
• alias labeled graph, alias transition system
• possible worlds
• alias nodes, alias states
• valuations
• alias interpretations
• links
• alias accessibility relations

p,q
p,q
p,q
p,q
5
Possible worlds models readings of R

• alethic
• Rwu iff u is possible, given the actual world w
• temporal
• Rwu iff u is in the future of w
• epistemic
• Riwu iff u is possible for agent i, given the
  actual world w
• deontic
• Rwu iff u is an ideal version of w
• dynamic
• Rawu iff u is a possible result of the
  execution of program/action a in w
• comparative (preferential, )
• Rwu iff w is smaller than u
• Rvwu iff w is smaller than u, given v
• reading of R ? properties of R

6
Possible worlds models properties of R

• only one relation
• serial forall w exists u Rwu
• reflexive
• transitive
• Euclidian
• confluent (Church-Rosser)
• dense
• well-founded (not FO-definable!)

• more than one
• R1 included in R2
• R1 R2?R3
• R2 (R1)-1 (transitive closure)
• R2 (R1) (transitive closure)
• R1 R2 R2 R1
• Church-Rosser

7
Language modal operators

• non-truth functional concepts
• belief, time, action, obligation, conditional,
• schema op(a1,,an)
• Beli A agent i believes that A
• F A A will be true at some future time
  point
• Aftera A A is true after action a
• Oblg A A is obligatory
• Oblgi A A is obligatory for agent i
• condC A A is true under hypothesis C (noted
  CgtA)
• generic form
• A A is necessary (true in all possible
  worlds)
• ltgtA A is possible
• in general A same as ltgtA
• except in substructural logics (intuitionistic, )

8
Interpreting the language truth conditions

• classical connectives
• M,w - P iff Vw(P) 1, for P in Atoms
• M,w - A?B iff (M,w - A and M,w - B)
• non-classical connectives
• interpretation via accessibility relation R
• the basic modal operators
• M,w - A iff forall u Rwu implies M,u -
  A
• M,w - ltgtA iff exists u Rwu and M,u - A

9
Examples of truth conditions

• multimodal operators
• M,w - iA iff forall u Riwu implies M,u -
  A
• M,w - ltgtiA iff
• relation algebra operators
• M,w - -1A iff forall u R-1wu implies M,u
  - A
• M,w - i ? jA iff forall u (Ri?Rj)wu implies
  M,u - A
• M,w - A iff forall u Rwu implies M,u -
  A

10
Examples of truth conditionstemporal operators
R
R

• branching time operators
• M,w - ?XA iff exists R in Paths(w) M,R(w)
  - A
• (Paths(w) the set of paths going through
  w)

11
Examples of truth conditionstemporal operators
R
R

• branching time operators
• M,w - ?XA iff exists R in Paths(w) M,R(w)
  - A
• (Paths(w) the set of paths going through
  w)
• M,w - ?ltgtA iff ?R in Paths(w) ?n M,Rn(w) -
  A

12
Examples of truth conditionstemporal operators
A
A
A
B

• binary temporal operators
• M,w - A Until B iff exists u Rwu and M,u
  - B and
• forall u (Rwu and Rvu implies M,u - A )
• M,w - A Since B iff
• M,w - ?(A Until B) iff forall R in Paths(w)

13
Examples of truth conditionsimplications

• intuitionistic implication
• M,w - A gt B iff forall u Rwu implies M,u
  - A B
• conditional operator
• M,w - A gt B iff forall u RAwu implies M,u
  - B
• relevant implication
• M,w - A gt B iff forall u,u
• Rwuu implies (M,u - A implies M,u - B)

14
Models

• model M (W,R,V)
• W nonempty set (possible worlds)
• R Ops (WxW) (accessibility relation)
• V W (Atoms 0,1) (valuation)
• pointed model ((W,R,V),w)
• w in W (actual world)
• extension of A in M
• AM w in W M,w - A

15
Validity and satisfiability

• K the set of all models (Kripke)
• A is valid in K iff AM W for all M in K
  (K A)
• examples (P v P)
• (P?Q) P?Q
• P?Q (P?Q)
• A is satisfiable in K iff AM nonempy for some
  M in K
• examples P
• PÙP
• PÙP
• PÙP

16
Validity and satisfiability in a class of
models Cls

• Cls some subset of K
• A is valid in Cls iff AM W for all M in Cls
  (Cls A)
• examples P P invalid in K
• P P valid in the class of reflexive
  models
• ltgtP ltgtltgtP valid in transitive models
• A is satisfiable in Cls iff AM nonempy for
  some M in Cls
• examples PÙP satisfiable in K
• PÙP unsatisfiable in reflexive models
• A is valid in Cls iff A is unsatisfiable in Cls

17
Classes of models examples

• M card(W) 1
• Cls ltgtA A
• M card(W) 2
• Cls ltgt(AÙB) Ù ltgt(AÙB) B
• M card(W) finite
• M R() reflexive KT
• KT A A
• M R() transitive K4
• K4 ltgtltgtA ltgtA
• M R() equivalence relation S5
• S5 A ltgtA

18
Reasoning problems

• model checking
• given A, M and w, do we have M,w - A?
• validity
• given A and Cls, is A valid in Cls?
• satisfiability
• given A and Cls, does there exist M in Cls and w
  in M such that M,w - A?

19
Reasoning problems

• model checking
• given A, M and w, do we have M,w - A?
• validity
• given A and Cls, is A valid in Cls?
• satisfiability
• given A and Cls, does there exist M in Cls and w
  in M such that M,w - A?
• How can we solve them automatically?

20
Overview

• possible worlds semantics quickstart
• tableaux systems basic ideas
• tableaux systems basic definitions
• tableaux for simple modal logics
• tableaux for transitive modal logics
• tableaux for intuitionistic logic
• tableaux for other nonclassical logics
• tableaux for modal logics with transitive closure
  and other modal and description logics
• tableaux for 1st order logic
• some implemented tableaux theorem provers

21
The basic ideafor classical logic Beth

• try to find M and w by applying the truth
  conditions (tableau rules)
• M,w - AÙB ? add M,w - A, and add M,w - B
• M,w - A v B ? add either M,w - A, or add
  M,w - B
• (nondet.)
• M,w - A ? dont add M,w - A ???
• M,w - A ? add M,w - A
• M,w - (A v B) ? add M,w - A, and add M,w
  - B
• M,w - (AÙB) ? add either M,w - A, or add
  M,w - B
• apply while possible (downwards saturation")
• is this a model?
• NO if both M,w - P and M,w - P (tableau is
  closed)
• ELSE for every w, if M,w - P put Vw(P) 1,
  else put Vw(P) 0

22
The basic idea for modal logics

• apply truth conditions build a graph
• create nodes
• add links between nodes
• add formulas to nodes
• the basic cases
• M,w - A ? forall u such that Rwu, add u
  - A
• M,w - ltgtA ? add some new u, add Rwu, add u
  - A
• M,w - A ? add some new u, add Rwu, add u
  - A
• M,w - ltgtA ?
• downwards saturated graph is this a model?

23
The basic ideaexample for modal logic

• A P Ù P
• applying tableau rules
• M,w - PÙP

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The basic ideaexample for modal logic

• A P Ù P
• applying tableau rules
• M,w - PÙP
• M,w - PÙP, M,w - P, M,w - P

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The basic ideaexample for modal logic

• A P Ù P
• applying tableau rules
• M,w - PÙP
• M,w - PÙP, M,w - P, M,w - P

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The basic ideaexample for modal logic

• A P Ù P
• applying tableau rules
• M,w - PÙP
• M,w - PÙP, M,w - P, M,w - P
• M,w - PÙP, M,w - P, M,w - P, Rwu, M,u
  - P

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The basic ideaexample for modal logic

• A P Ù P
• applying tableau rules
• M,w - PÙP
• M,w - PÙP, M,w - P, M,w - P
• M,w - PÙP, M,w - P, M,w - P, Rwu, M,u
  - P
• no more tableau rule applies

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The basic ideaexample for modal logic

• A P Ù P
• applying tableau rules
• M,w - PÙP
• M,w - PÙP, M,w - P, M,w - P
• M,w - PÙP, M,w - P, M,w - P, Rwu, M,u
  - P
• no more tableau rule applies
• never both w - A and w - A (open tableau)
• ? model can be built M (W,R,V)
• set of worlds W W w,u
• accessibility relation R Rwu
• valuation V Vw(P) 1, Vu(P) 0

29
The basic idea example for modal logic (ctd.)

• premodel for
• A P Ù P
• ? not closed
• ? is a model of A

30
Historical remarks

• the early days (1950-80) handwritten proofs
• Beth, Gentzen
• relation to sequent calculus
• tableau proof sequent proof backwards
• Kripke explicit accessibility relation
• Smullyan, Fitting uniform notation
• today mechanized systems
• fast provers exist
• FaCT Horrocks
• K-SAT GiunchigliaSebastiani
• importance of strategies
• applications exist BDI logics, description logics

31
Overview

• possible worlds semantics quickstart
• tableaux systems basic ideas
• tableaux systems basic definitions
• tableaux for simple modal logics
• tableaux for transitive modal logics
• tableaux for intuitionistic logic
• tableaux for other nonclassical logics
• tableaux for modal logics with transitive closure
  and other modal and description logics
• tableaux for 1st order logic
• some implemented tableaux theorem provers

32
Informal definition of tableau rules

• Tableau rules expand directed graphs by
• adding formulas
• adding nodes
• adding links
• duplicating the graph
• rule(G) G1,,Gn
• rule(G1,,Gn) rule(G1)??rule(Gn)

33
Informal definition of tableau rules

• Tableau rules expand directed graphs by
• adding formulas
• adding nodes
• adding links
• duplicating the graph
• rule(G) G1,,Gn
• rule(G1,,Gn) rule(G1)??rule(Gn)
• LoTREC apply rule to G
• apply rule to every formula in every node of G
• ? strategies get more declarative ? proofs get
  easier

34
Tableau rules syntax

• general form
• rule ruleName
• if cond1
• if condn
• do action1
• do actionk

• .

35
Tableau rules syntax

• general form
• rule ruleName
• if cond1
• if condn
• do action1
• do actionk

• example conditions
• if hasElement node formula
• if isLinked node1 node2 R
• ... (more to come)
• example actions
• do stop
• do addElement node formula
• do newNode node
• do link node1 node2 R
• do duplicate node1
• ... (more to come)

36
Example tableau rules for classical logic

• the
• LoTREC
• tableau
• prover

37
Example tableau rules for classical logic

• declaration of connectors
• negation and conjunction only

38
Example tableau rules for classical logic

• rule Stop
• if there is an explicit contradiction
• then stop exploring the tableau

39
Example tableau rules for classical logic

• rule NotNot
• replaces A by A

40
Example tableau rules for classical logic

• rule And
• if A B is in a node
• then add A and B to node

41
Example tableau rules for classical logic

• rule NotAnd
• if (AB) is in a node
• then duplicate tableau,
• add A to the first tableau

42
Definition of strategies

• A strategy defines some order of application of
  the tableau rules
• firstrule rule1 rulen end
• apply first applicable rule (and stop)
• allrules rule1 rulen end
• apply all applicable rules in order
• repeat strategy end
• repeat until no rule applicable
• Strategy stops if no rule is applicable.

43
Strategy for classical logic

• strategy CPLStrategy
• repeat allRules
• Stop
• NotNot
• And
• NotAnd
• end end
• end
• ? fair strategy

44
Strategy for classical logic example

• CPLStrategy(P(PQ))

45
Strategy for classical logic example (ctd.)

• CPLStrategy(P(PQ))
• T1 , T2

46
Definition of tableaux

• The set of tableaux for A with strategy S is
• the set of graphs
• obtained by applying the strategy S
• to an initial single-node graph
• whose root contains only A.
• notation S(A)
• Remark
• our tableau tableau branch in the literature
• (sounds odd to call a graph a branch)

47
Tableaux open or closed?

• A node is closed iff it contains FALSE.
• A tableau is closed iff it has a closed node.
• A set of tableaux is closed
• iff all its elements are.
• An open tableau is a premodel
• ? build a model

48
Formal properties

• to be proved for each strategy
• Termination
• For every A, S(A) terminates.
• Soundness
• If S(A) is closed then A is unsatisfiable.
• Completeness
• If S(A) is open then A is satisfiable.

49
In general

• soundness proofs easy (we apply truth
  conditions)
• termination proofs not so easy (case-by-case)
• completeness proofs
• for fair strategies
• standard techniques, work in most cases
• but fair strategies do not terminate in general
• for terminating strategies
• difficult (rigorous proofs rare even for the
  basic modal logics!)
• reason strategy imperative programming
• but soundness termination is practically
  sufficient (e.g. when experimenting with a
  logic)
• apply strategy to A
• take an open tableau and build pointed model
  (M,w)
• check if M in model class
• check if M,w - A

50
A general termination theorem

• IF for every rule r
• (the do-part of r only contains strict
• subformulas of the if-part of r
• AND
• some restriction on node creation)
• THEN
• for every formula A
• the tableaux construction terminates

51
Another general termination theorem

• IF for every rule r
• (the do-part of r only contains
• subformulas of the if-part of r
• AND
• some restriction on node creation)
• AND
• some looptesting in the strategy
• THEN
• for every formula A
• the tableaux construction terminates

52
Overview

• possible worlds semantics quickstart
• tableaux systems basic ideas
• tableaux systems basic definitions
• tableaux for simple modal logics
• tableaux for transitive modal logics
• tableaux for intuitionistic logic
• tableaux for other nonclassical logics
• tableaux for modal logics with transitive closure
  and other modal and description logics
• tableaux for 1st order logic
• some implemented tableaux theorem provers

53
The basic modal logic K

• the basic modal operators
• M,w - A iff forall u Rwu implies M,u -
  A
• M,w - ltgtA iff exists u Rwu and M,u - A

54
Tableau rules for K

• connectors not, and, nec
• some rules for classical logic

55
Tableau rules for K

• connectors not, and, nec
• some rules for classical logic
• createSuccessor
• if not nec A is in node0
• then create new node node1
• link it to node0
• add not A to node1
• end

56
Tableau rules for K

• connectors not, and, nec
• some rules for classical logic
• propagateNec
• if nec A is in node0
• node0 is linkednode1 R
• then add node1 A
• end

57
Tableaux for K

• plus rules for the definable connectives
• KStrategy(ltgtP ltgtQ (R v ltgtS))

58
Termination of KStrategy

• For every A, KStrategy(A) terminates.
• Proof
• Every tableau rule only adds strict subformulas.
• The createSuccessor rule is only applied once per
  occurrence of A.
• This can only be done a finite number of times,
  then the strategy stops.
• ? follows from general termination theorem

59
Soundness

• If KStrategy(A) is closed then A is
  unsatisfiable.
• Proof
• Every tableau rule is guaranteed by the truth
  conditions
• If G is K-satisfiable
• then there is Gi in rule(G) that is K-satisfiable
• Hence if every graph is closed
• then the original A cannot be satisfiable.

60
Completeness

• If KStrategy(A) is open then A is satisfiable.
• Proof
• Take some open tableau G in KStrategy(A).

61
Completeness

• If KStrategy(A) is open then A is satisfiable.
• Proof
• Take some open tableau G in KStrategy(A).
• G is a downwards closed set (Hintikka set)
• if A in node then A in node
• if AB in node then A in node and B in node
• if (AB) in node then A in node or B in node
• if A in node then there is an accessible node
  containing A
• if A in node then every successor node
  contains A
• (because allRules strategy is fair every rule
  eventually applies)

62
Completeness

• If KStrategy(A) is open then A is satisfiable.
• Proof
• Take some open tableau G in KStrategy(A).
• G is a downwards closed set (Hintikka set)
• if A in node then A in node
• if AB in node then A in node and B in node
• if (AB) in node then A in node or B in node
• if A in node then there is an accessible node
  containing A
• if A in node then every successor node
  contains A
• (because allRules strategy is fair every rule
  eventually applies)
• Build a K-model from G
• Vnode(P) 1 iff P appears in node

63
Completeness

• If KStrategy(A) is open then A is satisfiable.
• Proof
• Take some open tableau G in KStrategy(A).
• G is a downwards closed set (Hintikka set)
• if A in node then A in node
• if AB in node then A in node and B in node
• if (AB) in node then A in node or B in node
• if A in node then there is an accessible node
  containing A
• if A in node then every successor node
  contains A
• (because allRules strategy is fair every rule
  eventually applies)
• Build a K-model from G
• Vnode(P) 1 iff P appears in node and P atomic
  
• Prove by induction on the form of A
• for every A in node, M,node - A
• (fundamental lemma)

64
Modal logic KT

• accessibility relation is reflexive
• idea integrate this into truth condition
• M,w - A iff forall u Rwu implies M,u - A
  

65
Modal logic KT

• accessibility relation is reflexive
• idea integrate this into truth condition
• M,w - A iff forall u Rwu implies M,u - A
  
• and M,w - A

66
Tableauxfor modal logic KT

• connectors as for K
• rules as for K

67
Tableauxfor modal logic KT

• connectors as for K
• rules as for K
• plus when A is in a node then add A to it
• KTStrategy(P P)

68
Tableauxfor modal logic S5

• accessibility relation is equivalence relation
• can be supposed to be a single equivalence class
• optimized tableau rules

69
Overview

• possible worlds semantics quickstart
• tableaux systems basic ideas
• tableaux systems basic definitions
• tableaux for simple modal logics
• tableaux for transitive modal logics
• tableaux for intuitionistic logic
• tableaux for other nonclassical logics
• tableaux for modal logics with transitive closure
  and other modal and description logics
• tableaux for 1st order logic
• some implemented tableaux theorem provers

70
Tableau rules for S4

• accessibility relation is reflexive and
  transitive
• tableau rules for S4
• connectors as for KT
• rules as for KT
• and take into account transitivity
• when A is in a node
• then add A to all children

71
Tableau rules for S4

• accessibility relation is reflexive and
  transitive
• tableau rules for S4
• connectors as for KT
• rules as for KT
• and take into account transitivity
• if A is in a node
• then add A to all children
• problem find a terminating strategy

72
Tableau rules for S4

• Example M,w - P
• add M,w - P (by rule for reflexivity)

73
Tableau rules for S4

• Example M,w - P
• add M,w - P (by rule for reflexivity)
• create u, add Rwu, add M,u - P
• (by createSuccessor)

74
Tableau rules for S4

• Example M,w - P
• add M,w - P (by rule for reflexivity)
• create u, add Rwu, add M,u - P
• (by createSuccessor)
• add M,u - P (by rule for transitivity)

75
Tableau rules for S4

• Example w - P
• add M,w - P (by rule for reflexivity)
• create u, add Rwu, add M,u - P
• (by createSuccessor)
• add M,u - P (by rule for transitivity)
• add M,u - P (by rule for reflexivity)

76
Tableau rules for S4

• Example w - P
• add w - P (by rule for reflexivity)
• create u, add Rwu, add u - P
• (by createSuccessor)
• add M,u - P (by rule for transitivity)
• add M,u - P (by rule for reflexivity)
• create u

77
Tableau rules for S4

• Example w - P
• add M,w - P (by rule for reflexivity)
• create M,u, add Rwu, add M,u - P
• (by createSuccessor)
• add M,u - P (by rule for transitivity)
• add M,u - P (by rule for reflexivity)
• create u
• put a looptest into the rules!

78
Tableau rules for S4 (ctd.)

• principle
• if a node is included in an ancestor
• then mark it.

79
Tableau rules for S4 (ctd.)

• principle
• if a node is included in an ancestor
• then mark it.
• if a node is marked
• then block the createSuccessor rule
• S4Strategy(P)

80
S4Strategy(ltgt (P v Q) ltgtP ltgtQ)
81
Overview

• possible worlds semantics quickstart
• tableaux systems basic ideas
• tableaux systems basic definitions
• tableaux for simple modal logics
• tableaux for transitive modal logics
• tableaux for intuitionistic logic
• tableaux for other nonclassical logics
• tableaux for modal logics with transitive closure
  and other modal and description logics
• tableaux for 1st order logic
• some implemented tableaux theorem provers

82
Intuitionistic logic

• no modal operators, but different semantics for
  implication and negation
• aim invalidate
• (PgtFALSE) gt P ex falso quodlibet
• P v P tertio non datur
• (Q gt P) gt (P gt Q) contraposition
• R is reflexive, transitive and hereditary
• if Rwu and Vw(P) 1 then Vu(P) 1
• similar to S4
• truth condition
• M,w - AgtB iff forall u Rwu implies M,u -
  A?B

83
Tableaux rules for intuitionistic logic

• follow translation from LJ to S4
• P P (inheritance)
• (AgtB) (A B)
• (A) (A)
• tableaux similar to S4
• signed formulas
• T(P) P is true
• F(P) P is false
• F(P) ? P

84
Tableaux rules for intuitionistic logic

• create successor
• make AgtB false in w
• create u, add link Rwu,
• make A false in u,
• make B true in u

85
Tableaux rules for intuitionistic logic

• create successor
• make AgtB false in w
• create u, add link Rwu,
• make A false in u,
• make B true in u
• inheritance
• if M,w - P and Rwu
• then add M,u - P

86
Tableaux rules for intuitionistic logic PgtP
LJStrategy(((PgtFalse)gtFalse)gtP) ? 4
tableaux, 1 open
87
Overview

• possible worlds semantics quickstart
• tableaux systems basic ideas
• tableaux systems basic definitions
• tableaux for simple modal logics
• tableaux for transitive modal logics
• tableaux for intuitionistic logic
• tableaux for other nonclassical logics
• tableaux for modal logics with transitive closure
  and other modal and description logics
• tableaux for 1st order logic
• some implemented tableaux theorem provers

88
Relevant logics

89
Paraconsistent logics

90
Overview

• possible worlds semantics quickstart
• tableaux systems basic ideas
• tableaux systems basic definitions
• tableaux for simple modal logics
• tableaux for transitive modal logics
• tableaux for intuitionistic logic
• tableaux for other nonclassical logics
• tableaux for modal logics with transitive closure
  and other modal and description logics
• tableaux for 1st order logic
• some implemented tableaux theorem provers

91
Linear Temporal Logic

• two modal operators
• always
• X next
• R(X) is serial and deterministic
• R() R(X))
• R() linear order
• mix axioms
• A ? AÙXA
• ltgtA ? A v XltgtA
• induction axiom
• AÙ(AXA) A
• decidable, EXPTIME complete

92
Tableau rules for Linear Temporal Logic

• how take induction into account?
• solution dont care, and only apply the mix
  axioms
• rewrite A to A Ù XA
• rewrite ltgtA to A v XltgtA
• only create successors for X, never for ltgt
• termination use the looptest from transitive
  modal logics
• nodes only contain subformulas of orig. formula
• looptest succeeds at most at polynomial depth

93
Tableau rules for Linear Temporal Logic example

• Example M,w - P
• add M,w - PÙXP (by mix axioms)
• add M,w - P, M,w - XP
• create u, add RXwu, add M,u - P
• (by propagation rule for X)
• add M,u - PÙXP (by mix axioms)
• add M,u - P, M,u - XP
• w contains u mark u contained

94
Tableau rules for Linear Temporal Logic (ctd.)
P
P
P
P

• may result in nonstandard models of ltgtP
• ? P never fulfilled
• ? check if all ltgt are fulfilled!

95
Tableau rules for Linear Temporal Logic example

• Example LTLStrategy(ltgtP)
• M,w - ltgtP
• .
• .

96
Tableau rules for Linear Temporal Logic

• Example LTLStrategy(ltgtP)
• M,w - ltgtP
• M,w - P v XltgtP (by mix)
• .

97
Tableau rules for Linear Temporal Logic

• Example LTLStrategy(ltgtP)
• M,w - ltgtP
• M,w - P v XltgtP (by mix)
• M1,w - ltgtP, M1,w - P M2,w - ltgtP, M2,w -
  XltgtP
• (nothing applies)
• .

98
Tableau rules for Linear Temporal Logic

• Example LTLStrategy(ltgtP)
• M,w - ltgtP
• M,w - P v XltgtP (by mix)
• M1,w - ltgtP, M1,w - P M2,w - ltgtP, M2,w -
  XltgtP
• (nothing applies) RXwu, M2,u - ltgtP
• .

99
Tableau rules for Linear Temporal Logic

• Example LTLStrategy(ltgtP)
• M,w - ltgtP
• M,w - P v XltgtP (by mix)
• M1,w - ltgtP, M1,w - P M2,w - ltgtP, M2,w -
  XltgtP
• (nothing applies) RXwu, M2,u - ltgtP
• M2,u - P v XltgtP (by mix)

100
Tableau rules for Linear Temporal Logic

• Example LTLStrategy(ltgtP)
• M,w - ltgtP
• M,w - P v XltgtP (by mix)
• M1,w - ltgtP, M1,w - P M2,w - ltgtP, M2,w -
  XltgtP
• (nothing applies) RXwu, M2,u - ltgtP
• M2,u - P v XltgtP (by mix)
• M21,u - P M22,u - XltgtP

101
Tableau rules for Linear Temporal Logic

• Example LTLStrategy(ltgtP)
• M,w - ltgtP
• M,w - P v XltgtP (by mix)
• M1,w - ltgtP, M1,w - P M2,w - ltgtP, M2,w -
  XltgtP
• (nothing applies) RXwu, M2,u - ltgtP
• M2,u - P v XltgtP (by mix)
• M21,u - P M22,u - XltgtP
• (nothing applies) contained in w

102
Tableau rules for Linear Temporal Logic

• Example LTLStrategy(ltgtP)
• M,w - ltgtP
• M,w - P v XltgtP (by mix)
• M1,w - ltgtP, M1,w - P M2,w - ltgtP, M2,w -
  XltgtP
• (nothing applies) RXwu, M2,u - ltgtP
• M2,u - P v XltgtP (by mix)
• M21,u - P M22,u - XltgtP
• (nothing applies) u contained in w
• ltgtP not fulfilled

103
Propositional dynamic logic (PDL)

• two kinds of expressions
• formulas
• A P A A?B pA
• programs
• p a p1p2 p1?p2 p A?
• in the models R interprets programs
• R(p1p2) R(p1)R(p2)
• R(p1?p2) R(p1)?R(p2)
• R(p) (R(p))
• R(A?) ltw,wgt M,w - A

104
Tableaux for PDL

• similar to LTL
• expand pA to A ? ppA
• dont apply createSuccessor to formulas pA
• mark nodes that are included in some ancestor
• dont apply createSuccessor to formulas pA if
  node is marked
• expand the other program expressions
• p1p2A ? p1p2A
• p1?p2A ? p1A ? p2A
• A?B ? AB

105
Description logics

• roles and concepts
• more expressive than classical propositional
  logic
• less expressive than 1st order logic
• focus on decidable logics
• applications
• databases
• software engineering
• web-based information systems
• description of medical terminology
• ontology of the semantic web
• standards DAMLOIL, OWL
• description of web services
• WSDL, OWL-S

106
Description logicsconcepts and roles

• roles binary relations
• hasChild
• hasHusband
• concepts unary relations properties
• Person
• Female
• Parent n Female
• Father U Mother
• Parent
• ?hasChild.Female individuals having a female
  child
• ?hasChild.Female
• gt1 hasChild.T individuals having more than 1
  child
• set of concepts ? assertion box (ABox)

107
Description logics TBoxes

• set of relations between concepts and roles
• ? terminological box (TBox)
• restricted to concept abbreviations (sometimes
  fixpoint definitions)
• Mother Person n Female
• are expanded away ? TBox ?

108
Description logicsreasoning tasks

• satisfiability of a concept C
• subsumption of C1 by C2
• same as C1n C2 unsatisfiable
• equivalence of C1 by C2
• same as C1 subsumes C2 and C1 subsumes C2
• disjointness of C1 and C2
• ? subsumes C1nC2
• all reasoning tasks reduce
• to concept satisfiability

109
Description logics

• translation of concepts into modal logics
• ?hasChild.Female lthasChildgtFemale
• ?hasChild.Female hasChild.Female
• Parent n Female Parent ? Female
• Father U Mother Father v Mother
• lt2 hasChild.T hasChild2 T
• 2 hasChild.T lthasChildgt2 T
• modal logics with number restrictions
  FattorosiBarnaba, van der Hoek

110
Description logics

• description logic ALC
• C
• C1 n C2
• C1 U C2
• ?R.C
• ?R.C
• multimodal K
• description logic ALCreg
• ALC regular expressions on roles
• PDL
• all description logic reasoning tasks reduce
• to satisfiability checking in modal logics
• tableaux used as optimal decision procedures

111
Logics of action and knowledge

• 2 modal operators
• Knwi A agent i knows that A
• a A after execution of action a, A holds
• product logics
• RKnwiRa RaRKnwi (permutation)
• if wRKnwiu and wRav then exists t such that
  uRat and vRKnwit
• (confluence)
• axiomatically
• KnwiaA ? aKnwiA
• ltagtKnwiA KnwiltagtA
• tableaux
• ? problem combination with transitivity

112
Belief-Desire-Intention logics

• Bratman, RaoGeorgeff
• 3 modal operators
• Beli A agent i believes that A
• Desirei A agent i desires that A
• Intendi A agent i intends that A
• plus branching time logic

113
Modal logics with density

• accessibility relation is dense
• if Rwu then exists v Rwv and Rvu

114
Non-normal modal logics

• no accessibility relation, but neighborhood
  functions N W 22W
• M,w - A iff exists U in N(w) forall u in U
  M,u - A
• non-normal modal logic EM
• can be represented by a set of relations
• M,w - A iff exist Ri forall u (Riwu implies
  M,u - A)
• logic EM non-normal
• not valid P?Q (P?Q)
• but valid (P?Q) P?Q

115
Tableau rules for EM

116
Overview

• possible worlds semantics quickstart
• tableaux systems basic ideas
• tableaux systems basic definitions
• tableaux for simple modal logics
• tableaux for transitive modal logics
• tableaux for intuitionistic logic
• tableaux for other nonclassical logics
• tableaux for modal logics with transitive closure
  and other modal and description logics
• tableaux for 1st order logic
• some implemented tableaux theorem provers

117
1st order logic

• How should we handle the quantifiers?
• ?x p(x) ? p(a) is unsatisfiable
• ?x p(x) ? ?x p(x) is unsatisfiable
• naïve implementation Beth, Smullyan
• if hasElement node0 forall x A(x)
• do createTerm t (doesnt exist in LoTREC yet)
• do add node0 A(t)
• if hasElement node exists x A(x)
• do createNewConstant c
• do add node A(c)
• problem loops for satisfiable formulas

118
Herbrand Tableaux for 1st order logic

• 1st solution restrict instantiation to Herbrand
  universe
• if hasElement node0 forall x A(x)
• do createHerbrandTerm t (doesnt exist in
  LoTREC yet)
• do add node0 A(t)
• ex. ?x p(x,x) ? ?x?y p(x,y)) satisfiable
• 1. ?x p(x,x)
• 2. ?x?y p(x,y)
• 3. ?y p(a,y) (2), new constant
• 4. p(a,a) (3), only Herbrand term
• 5. p(b,b) (1), new constant
• 6. p(a,b) (3), Herbrand term
• no further instantiation of (3) is possible
• decision procedure for formulas without positive
  ??

119
Herbrand Tableaux for 1st order logic

• counterexample ?x?y p(x,y) satisfiable
• 1. ?x?y p(x,y)
• 2. ?y p(a,y) (1), Herbrand term
• 3. p(a,b) (2), new constant
• 4. ?y p(b,y) (1), Herbrand term
• 5. p(b,c) (4), new constant
• 6.
• ? loops

120
Free-variable tableaux with unification

• 2nd solution dont instantiate at all
• work with free variables
• runtime skolemization of existential quantifiers
• term unification
• ex. ?x?y p(x,y) ? ?x?y p(x,y)) satisfiable
• 1. ?x?y p(x,y)
• 2. ?x?y p(x,y)
• 3. ?y p(x1,y) from (1), replace x by free x1
• 4. ?y p(x2,y) from (2), replace x by free x2
• 5. p(x1,f(x1)) from (3), Skolem function f(x1)
• 6. p(x2,g(x2)) from (4), Skolem function g(x2)
• stops (5) and (6) dont unify
• but doesnt terminate in all cases (sure)
• else 1st order logic would be decidable

121
Overview

• possible worlds semantics quickstart
• tableaux systems basic ideas
• tableaux systems basic definitions
• tableaux for simple modal logics
• tableaux for transitive modal logics
• tableaux for intuitionistic logic
• tableaux for other nonclassical logics
• tableaux for modal logics with transitive closure
  and other modal and description logics
• tableaux for 1st order logic
• some implemented tableaux theorem provers

122
LoTREC

• IRIT-CNRS Toulouse (Sahade, Gasquet, Herzig)
  accessible through www
• general theorem prover
• explicit accessibility relations
• easy to implement logics with symmetric
  accessibility relations etc.
• back-and-forth rules
• inefficient

123
TableauxWorkBench (TWB)

• Australian National U. (Abate, Goré)
• general theorem prover
• close to Gentzen sequents
• accessibility relations remain implicit
• hard to implement logics with symmetric
  accessibility relations
• temporal logic with future and past
• converse of programs

124
LogicWorkBench (LWB)

• U. Bern (Jäger, Heuerding) accessible through
  www
• efficient algorithms for all the basic modal and
  temporal logics
• hard to implement a new logic

125
FaCT

• U. Manchester (Horrocks) open source
• fast decision procedure for description logics
  with inverse roles and qualified number
  restrictions
• multimodal K converse number restrictions
• optimized backtracking backjumping

126
KSAT

• U. Trento (Giunchiglia, Sebastiani)
• combines tableaux method with fast SAT solvers
  for classical propositional logic
• call a SAT solver, where subformulas A, ltgtB are
  viewed as atomic
• SAT solver returns a tentative valuation
• use modal tableau rules to generate children
• if inconsistent then there is no model
• else iterate
• very efficient
• exists for all basic modal logics

127
KSAT (ctd.)

• KSAT((PQ) ltgtP)
• call SAT with set of clauses (PQ), ltgtP
• SAT returns
• V((PQ)) 1
• V(ltgtP) 1
• apply createOneSuccessor and propagateNec
• w - (PQ), w - ltgtP, Rwu, u - P, u -
  PQ
• call SAT with set of clauses P,Q,P
• SAT returns
• set of clauses unsatisfiable
• (PQ) ltgtP is unsatisfiable in K

128
Conclusion

• search for models exploit the truth conditions
• tableaux work both ways
• finding a model
• refuting
• termination decidability
• tableaux as optimal decision procedures
• ? description logics

129
Thanks to

• Mohamad Sahade
• Olivier Gasquet
• Luis Fariñas del Cerro
• Dominique Longin
• Tiago Santos de Lima
• Fabrice Evrard
• Carole Adam
• Nicolas Troquard
• Benoit Gaudou
• Ivan Varzinczak
• Bilal Saïd
• Dominique Ziegelmayer
• and the other members of the LILaC group